Preheat the oven to 350 degrees F.
Bring 8 quarts water to a rolling boil and add 2 tablespoons salt.
Roughly chop the Swiss chard, discarding the rough stems. Add the Swiss chard to the boiling water and cook until tender, about 15 minutes. Drain thoroughly and set aside.
In a 12-inch saucepan, heat 3 tablespoons olive oil over a medium flame until hot but not smoking. Add the onion and garlic, and cook until soft and golden brown, about 5 minutes. Add the Swiss chard and the parsley. Let cook over medium heat for about 10 minutes, stirring occasionally. Remove from the heat and let cool.
Meanwhile, break the eggs into a small bowl. Season, to taste, with salt and pepper. Add 3 tablespoons of Parmigiano and, using a whisk, mix until the ingredients are well-blended. Add the egg mixture to the cooled Swiss chard and toss to combine.
Using the remaining olive oil to lightly grease a shallow 9-inch round or oval baking dish. Dust the bottom of the baking dish with 1/2 cup bread crumbs. Carefully place the Swiss chard and egg mixture into the pan. Dust with the remaining Parmigiano and then the remaining bread crumbs.
Bake until the top is golden brown, about 1 hour. Serve hot or room temperature.